Improved frictional modeling for the pressure-time method

Flow Measurement and Instrumentation 69 (2019) 101604

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Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst

Improved frictional modeling for the pressure-time method a,*

L.R. Joel Sundstrom , Simindokht Saemi a b c

a,b

c

, Mehrdad Raisee , Michel J. Cervantes

T a

Luleå University of Technology, 971 87 Luleå, Sweden School of Mechanical Engineering, College of Engineering, University of Tehran, P. O. Box: 11155-4563, Tehran, Iran Hydraulic Machinery Research Institute, School of Mechanical Engineering, College of Engineering, University of Tehran, P. O. Box: 11155-4563, Tehran, Iran

A R T I C LE I N FO

A B S T R A C T

Keywords: Pressure-time method Transient friction modeling Pipe flow

The pressure-time method is classified as a primary method for measuring discharge in hydraulic machinery. The uncertainty in the discharge determined using the pressure-time method is typically around ± 1.5 %; however, despite dating back almost one hundred years in time, there still exists potential to reduce this uncertainty. In this paper, an improvement of the pressure-time method is suggested by implementing a novel formulation to model the frictional losses arising in the evaluation procedure. By analyzing previously obtained data from CFD, laboratory and full-scale pressure-time measurements it is shown that the new friction model improves the accuracy of the flow rate calculation by approximately 0.1–0.2% points, compared to currently utilized friction models. Despite being a small absolute improvement, the new friction model presents an important development of the pressure-time method because the relative improvement is significant.

2010 MSC: 00-01 99-00

1. Introduction Assessing the wall shear stress is of fundamental importance in studies of canonical wall-bounded turbulent flows because it defines the friction velocity u τ = τ / ρ , used for scaling of mean and turbulent quantities (Monty et al. [1], e.g.). τ and ρ denote, respectively, the wall shear stress and the fluid's density. Assessing the wall shear stress is also of importance in technical applications, such as for determining the flow rate in hydraulic machinery or for detecting leakages in pipelines (see Jonsson et al. [2] or Colombo et al. [3]). The present paper concerns the mathematical modeling of wall friction with application to flow rate measurements in hydraulic machinery. The pressure-time method is a cost-efficient way to measure hydraulic turbine flow rate. The underlying principle of the method is to subject a statistically steady flow to either an acceleration or a deceleration. An estimate of the differential change in flow rate, ΔQ , resulting from the transient can then be obtained by measuring the differential pressure over a distance L using

− ΔQ = Q0 − Qf =

A ρL

tu

∫ (Δp + ξ )dt, 0

(1)

where Q0 , Qf , A, tu , Δp and ξ represent the initial flow rate, the final flow rate, the cross-sectional area, the upper integration limit, the differential pressure and the frictional losses, respectively. When using the pressure-time method, the turbine is usually brought to a complete rest,

*

and for such occasions Qf represents the leakage flow through the agency that generated the deceleration, typically the turbine wicket gates or a valve. In this paper, the upcoming discussion is restricted to cases in which there is no final mean flow, except for eventual leakage flows. Note that the loss term is related to the wall shear stress through ξ = −4τL/ D , in which D denotes the pipe diameter. In a full-scale hydropower plant the differential pressure can be measured during a pressure-time measurement, whereas the wall shear stress cannot be measured. Therefore, the frictional losses have to be modeled in the evaluation procedure of a pressure-time measurement. In the IEC41 standard [4] (International standard - field acceptance tests to determine the hydraulic performance of hydraulic turbines, storage pumps and pump-turbines), ξ is recommended to be modeled using a constant loss-coefficient times the instantaneous flow rate squared. Despite the simplicity of this frictional formulation, Sundstrom & Cervantes [5] showed that the IEC41-modeled wall shear stress agrees well with measurements of τ for times prior to the complete blockage of the flow (tb ) in a pressure-time measurement. The data were obtained using hotfilm anemometry in a 300 mm internal diameter pipe at initial Reynolds numbers (Re = Ub D / ν , where Ub and ν denote the bulk velocity and the fluid's kinematic viscosity, respectively) of 1.7 × 106 and 0.7 × 106 , for closing times ranging between 4 s and 9 s. For times larger than tb , however, the IEC41-modeled and measured wall shear stresses did not agree. In the work by Jonsson et al. [2], a frictional formulation different

Corresponding author. E-mail addresses: [email protected] (L.R.J. Sundstrom), [email protected] (S. Saemi), [email protected] (M. Raisee), [email protected] (M.J. Cervantes).

https://doi.org/10.1016/j.flowmeasinst.2019.101604 Received 24 May 2018; Received in revised form 8 January 2019; Accepted 14 March 2019 Available online 01 August 2019 0955-5986/ © 2019 Elsevier Ltd. All rights reserved.

Flow Measurement and Instrumentation 69 (2019) 101604

L.R.J. Sundstrom, et al.

A two-dimensional axisymmetric unsteady CFD simulation employing a low-Reynolds-number k − ω SST turbulence model has been performed by Saemi et al. [10] using the commercial program Fluent. The near-wall flow was resolved up to y+ < 1, and a grid-independence study was performed to assure that results were independent of the computational mesh. The simulation is largely a numerical replicate of the NTNU laboratory set-up studied by Jonsson et al. [2]. In the simulation, the initial Reynolds number was 0.7 × 106 , and the 4.424 s closure of the gate was replicated using a dynamic moving mesh. The differential pressure trace and bulk velocity obtained from the CFD simulation agreed well with the experimental data, which thus verified that the numerical set-up was able to reproduce the integral flow quantities Δp , Ub and τ (the wall shear stress was not measured by Jonsson et al. [2], but 2τ / R = −ρdUb/dt − dp /dx , such that a good agreement of Δp and Ub implies a good agreement of the wall shear stress as well). The full-scale measurements were taken at the Porjus U9 power plant located in Sweden, see Jonsson & Cervantes [11]. The hydraulic head was 54 m with a maximum Reynolds number of approximately 14 × 106 . Three FP2000/FDW differential pressure sensors from Honeywell were used to perform measurements of Δp over a distance of 4.975 m within a straight pipe section, 1.8 m in diameter and 16 m long. The sensors have a measuring range of ± 0.5 bar with an accuracy of ± 0.25%. An eight path transit-time ultrasound flow meter measured the reference flow rate.

from that recommended in the IEC41 standard was introduced. Instead of using a constant loss-coefficient, they subdivided the frictional loss into a quasi-steady contribution and an unsteady contribution, i.e., ξ ~(fqs + fu ) Ub |Ub |. For the unsteady portion of the losses, use was made of a simplified version of the Brunone model [6]:

fu =

kD dUb , Ub |Ub | dt

(2)

in which k is a loss-coefficient that can be calculated using Vardy's shear decay coefficient (see Bergant et al. [7]). Originally, the Brunone model contains an advective term in conjunction to the temporal one. For pressure-time flow rate calculations in pipes with constant cross section area, however, the advective effects are negligible as shown by Jonsson [8], and can thus be ignored when calculating the unsteady friction. In conjunction to the comparison between the IEC41-modeled wall shear stress and the measured τ, Sundstrom & Cervantes [5] investigated the accuracy of the unsteady frictional formulation during a pressure-time measurement. Similarly to the results of the IEC41modeled wall shear stress, the unsteady-modeled wall shear stress agreed well with the measured τ for times prior to the complete blockage of the flow. For t > tb , however, the unsteady predictions disagreed appreciably from the measurements. From the foregoing discussion it appears that there exists a potential to improve the frictional modeling for t > tb , and thus in the end, the accuracy of measuring turbine flow rate using the pressure-time method. Since the time-interval tb < t < tu typically amounts to less than 10% of the entire closing sequence, the absolute improvement is expected to be small. However, as pointed out by Adamkowski & Janicki [9], improving the accuracy of the pressure-time method by as little as a few tenths of a percentage point still represents a significant merit, because every amelioration of the method is of importance. The merit of such (small) improvements is the motivation of the following work, namely, to improve the unsteady frictional formulation. The proposed model is valid during the later stages of a pressure-time measurement, and the new model is shown to improve the accuracy of flow rate estimations by 0.1–0.2 percentage points compared to currently used evaluation procedures.

3. New frictional formulation In the following sections, results of flow rate calculations as well as estimates of the wall shear stress obtained from the IEC41 formulation and the unsteady formulation are discussed. To simplify the upcoming notation, the IEC41 formulation is designated SP, shorthand for ‘standard pressure-time’, whereas the unsteady-friction-modeling approach introduced by Jonsson et al. [2] is designated UP; shorthand for ‘unsteady pressure-time’. In Jonsson et al. [2] and Jonsson & Cervantes [11] it was shown that the accuracy of pressure-time flow rate calculations are improved by using the UP approach instead of the SP approach. As fore mentioned, use was made of the Brunone friction factor to model the unsteady portion of the frictional losses. It does, however, exist several other unsteady friction models but these were not investigated by Jonsson and his co-workers. As such, a first step in developing the frictional modeling for pressure-time flow rate calculations is to investigate the performance of existing models to conclude which one that has the superior performance. Fig. 1(a) shows the time-development of τ during a pressure-time measurement obtained from CFD along with estimates using the unsteady friction models developed by Vardy & Brown [12], Brunone et al. [6], Zarzycki et al. [13], and Szymkiewick & Mitosek [14]. A close-up view showing the data for 4 s< t < 5 s is shown in Fig. 1(b). The time-development of the bulk velocity has been included in Fig. 1(a) and the vertical line at t = 4.424 s indicates the complete blockage of the flow. For t < 2 s the deceleration of the bulk velocity is small, thus implying that the unsteady portion of the wall shear stress is negligible. As a consequence, all frictional models agree well with the CFD results during these first 2 s. For t > 2 s, though, the deceleration rate increases and when the flow unsteadiness becomes stronger, the wall shear estimates due to Vardy & Brown and Zarzycki et al. start to deviate appreciably from the CFD data. These poor performances do most likely stem from the assumptions underpinning these models, namely, that the turbulent viscosity remains frozen during the deceleration event. For, as shown by Mathur [15] and Sundstrom & Cervantes [16], the assumption of a frozen turbulent viscosity following a linear deceleration is valid during a time interval of less than 200 vis2 cous time units (ν / u τ0 , with u τ0 being the friction velocity at the commencement of the deceleration). In comparison, each second of the

2. Computational and experimental set-ups The theoretical justification of the new frictional model is based on data from a computational fluid dynamics (CFD) study of the pressuretime method [10]. Furthermore, the performance of the new formulation for flow rate calculations using Eq. (1) is tested against this CFD data, as well as laboratory and full-scale measurements [5,11]. Before deriving the new model, these set-ups are briefly described in this section. For a fuller coverage, the reader should consult the publications referred to in this section. The laboratory measurements were performed in a large-scale laboratory facility located at the Norwegian University of Science and Technology (NTNU). The test section consisted of a 26.67 m long pipe of internal diameter 300 mm. Water was supplied to the test section from a 9.75 m high constant-head tank, and the highest achievable Reynolds number was 1.7 × 106 . By closing a hydraulically driven gate valve located at the end of the test section, pressure-time measurements could be imitated under controlled laboratory conditions. The set-up has been used for thorough investigations of the pressure-time method by Jonsson et al. [2] and Sundstrom & Cervantes [5]. In here, data from the latter study is used for verification of the new formulation. Measurements of the differential pressure was performed between 39D and 52D from the test section inlet using two UNIK 5000 absolute pressure sensors. The accuracy of these sensors are ± 0.04% of full-scale reading. As a reference for the pressure-time calculated flow rate, an electromagnetic flow meter was used to record the flow rate prevailing before the commencement of the pressure-time measurement at an accuracy of ± 0.3%. 2

Flow Measurement and Instrumentation 69 (2019) 101604

L.R.J. Sundstrom, et al.

Fig. 1. Time-developments of CFD and frictional modeling estimates of the wall shear stress during a pressure-time measurement. (a) Complete closure sequence; (b) close-up view for 4 s< t < 5 s .

pressure-time measurement plotted in Fig. 1 corresponds to approximately 10,000 viscous time units; therefore, the turbulent viscosity is expected to remain frozen only during a negligible portion of the current deceleration event. The models due to Brunone and Szymkiewick & Mitosek, on the other hand, perform well until t ≈ 4 s. From the closeup view presented in Fig. 1 (b), it is seen that the Brunone friction exhibits the best performance of all models except for t > 4.4 s where the Brunone estimate diverges rapidly from the CFD data. Results similar to those presented in Fig. 1 have been observed for measurement data as well, but these are not reported in here. From the foregoing evaluation of unsteady frictional models it is clear that the UP-estimated (i.e., Brunone model-estimated) wall shear stress is in good agreement with reference data for times prior to the complete gate closure whereas the agreement is poor for t > tb . It is therefore reasonable to focus the improvement of the frictional formulation to the times just prior to and after the complete gate closure. During the earlier times of a pressure-time measurement, there exists no apparent reason to discard the UP model. The approach of the novel frictional model will thus be as follows: the UP model is used until the wall shear stress fulfills a suitable criterion, which is chosen to be when τ = 0 . When this criterion is fulfilled, the UP formulation is exchanged for the new friction model, which is then used until the evaluation procedure has finished. The functional form of the new formulation is derived next. It should be noted that the frictional formulation derived in here is largely the same as those developed for laminar flows by Zielke [17] and Brereton [18]. The main novelty of the present work is how the formulation is used; i.e., in conjunction with another frictional model in a slowly decelerating turbulent flow during the evaluation procedure of a pressure-time measurement. Consider the equation governing the ensemble-averaged mean velocity, U (r , t ) , in a fully developed transient turbulent pipe flow

∂U 1 dp ν ∂ ⎛ ∂U ⎞ 1 ∂ = − + r − (ruv ) . ∂t ρ dx r ∂ r ∂ r r ∂ r ⎝  ⎠       

⏟ MI

PG

VF

TI

Fig. 2. Balancing terms in the mean momentum equation, Eq. (3), at t = 4.2 s.

only for y+ < 60 . The term demarcating Eq. (3) from a laminar formulation, namely, the TI, is negligible for y+ < 15 as well as for y+ > 600 (Re τ = u τ0 R/ ν ≈ 13,500 ). Since the TI is negligible over approximately 96% of the pipe radius, the first simplifying assumption in the derivation of the new friction model is to neglect the TI altogether for the times just prior to and after the complete blockage of the flow. The resulting formulation for U is purely viscous; hence, it is (in theory) possible to find an analytical solution for the evolution of the mean velocity with time. Finding a closed-form solution is, however, not straightforward unless the flow is steady or at rest at the instant from which the TI is neglected. For all practical cases, though, the velocity is identically zero only after the flow has come to a complete rest. To reconcile this problem, consider the time-development of τ plotted in Fig. 1(a). At t ≈ 4.2 s, the wall shear stress crosses zero; hence, τ takes the same value as it would have done if the ensemble-averaged mean velocity was identically zero at this instant. Since the solution to Eq. (3) becomes particularly simple if the initial velocity is zero, the second simplification that is imposed in the derivation of the new friction model is that the entire velocity field is zero when τ crosses zero. Because the calculations underlying the model require the time to start from zero, a new time variable t ′ = t − t τ = 0 ought to be introduced. After imposing the two simplifying assumptions, an analytical expression for the wall shear stress can be derived by applying a unilateral Laplace transform on Eq. (3). The Laplace-transformed velocity field reads

(3)

Following the notation by Chin et al. [19], the respective term is denoted mean inertia, MI, pressure gradient, PG, viscous force, VF, and turbulent inertia, TI. Wall-normal distributions of these quantities obtained from the CFD simulation are shown in Fig. 2 at t = 4.2 s, i.e., at a time just prior to the complete blockage of the flow. The friction velocity prevailing before the transient and the kinematic viscosity has been used for normalization of the terms in the mean momentum equation and of the wall-normal coordinate ( y+ = yu τ0 / ν ). Since the flow is fully developed, the PG is uniform in the wall-normal direction and it is of leading-order importance at all radii. The MI is also of leading-order importance at all wall-normal positions except in a thin layer next to the wall, for y+ < 5, in which it is an order of magnitude smaller than the PG. As expected, the direct effect of viscosity is restricted to the near-wall region, and consequently, the VF is important

( (

⎛J i Fˆ 0 Uˆ = ⎜ s ⎜J i 0 ⎝

s ν s ν

) − 1⎞⎟, ⎟ R) ⎠ r

(4)

where the hat denotes a transformed variable. In the derivation of Eq. (4), use has been made of the auxiliary conditions U (R, t ′) = 0 , and that the velocity is finite on the pipe axis. J0 is the Bessel function of the first 3

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kind of order zero, F = (1/ ρ)dp /dx whereas the imaginary unit and transform variable are denoted i and s, respectively. Differentiating Eq. (4) with respect to the radial coordinate and evaluating the result at the pipe wall gives the transform of the wall shear stress

sFˆ

τˆ = −ρR

(

s ν

sJ i

R

)

, (5)

with J (y ) = yJ0 (y )/ J1 (y ) being a quotient between Bessel functions of order zero and one. Note that the quotient (s / s ) has been introduced to avoid singularities in the solution. Since τˆ constitutes a product of the Laplace transforms of sFˆ and s−1/ J , the inverse transform (symbolically denoted L −1) is the convolution between the inverses of these two functions. The inverse of sFˆ is the time derivative of the pressure gradient plus an additional term related to the initial value of the pressure gradient, i.e., L −1 (sFˆ (s )) = dF /dt ′ + F (0) δD (t ′) , in which δD (t ′) denotes the Dirac delta function. Finding the inverse of s−1/ J , on the other hand, requires calculation. Formally, L −1 (s−1/ J ) can be found from the inversion formula using the residue theorem of complex analysis. The time-domain solution thus obtained consists of an absolutely convergent series of exponential functions. The convergence of this series is relatively slow as t ′ → 0 , though. Now, the time-development of the wall shear stress is sought only as t ′ → 0 (because the solution is used only during the late stage of a pressure-time measurement, which is short in terms of the time-variable t ′). A more fruitful approach, then, is to expand s−1/ J in a truncated series of negative rational powers of s as s → ∞ and invert this series term by term, since from the theory of Laplace transforms, this gives the time-domain behavior of the inverse transform as t ′ → 0 . Truncating the series expansion of s−1/ J after six terms yields the following inverse transform

⎛ L −1 ⎜

⎞ t ′1/2 ⎟ ≡ W (t ′) = α Γ(3 / 2) −

1

s ⎞ sJ ⎛i R ⎝ ⎝ ν ⎠⎠

−

α 4 t ′2 8 Γ(3)

−

25α5 t ′5/2 128 Γ(7 / 2)

−

α2 t′ 2 Γ(2)

3/2

−

α3 t′ 8 Γ(5 / 2)

52α 6 t ′3 . 128 Γ(4)

(6)

With Γ(⋅) being the gamma function, and α = ν / R2 . Retaining six terms in the expansion of W (t ′) is somewhat excessive because α is typically of order 10−2 − 10−3 s−1/2 , such that the higher-order terms are negligible. Actually, for t ′ < 0.01R2 / ν s, the leading-order term represents accurately the weighting function. Incorporating the weighting function in the convolution yields the final form of the time-domain representation of the wall shear stress. t′

∫ ddt * ⎡ ddpx (t′ − t *) ⎤ W (t *)dt * − R ddpx (0) W (t′).

τ (t ′) = −R

0

⎣

⎦

(7) Fig. 4. Flow rate estimation error.

Before evaluating the performance of the new frictional formulation in flow rate calculations, data of τ obtained from Eq. (7) during pressure-time flow rate excursions are compared with CFD and

Fig. 3. Evaluation of the NUP-estimated wall shear stress. (a) Comparison of NUP estimate with CFD. (b) Comparison of NUP estimate with measurements. 4

Flow Measurement and Instrumentation 69 (2019) 101604

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calculated and reference flow rates, respectively. For the experimental cases, the flow rate calculation errors are the mean values of 22 and 6 repetitions for the laboratory and full-scale tests, respectively, and the error bars denote plus minus one standard deviation. For the CFD case, the NUP predicts the flow rate with a 0.16% error, whereas the errors for the SP and UP approaches are 0.24% and 0.46%, respectively. For the laboratory cases, the error of the NUP is smaller than ± 0.5 % in each case, and the approach outperforms both the SP and UP methods three out of four times. The flow rate estimation error for the full-scale power plant data is larger than for the CFD and laboratory cases, and the maximum deviation of the NUP estimate from the reference is − 1 % (the maximum deviation of the SP approach is − 1.5 %). It is difficult to discern a difference in the performance of the UP and NUP methods, both outperforming the SP approach in each case. A plausible reason for the less satisfactory performance of the NUP relative to the UP for the full-scale cases might be because the acceleration rate is relatively small just prior to the complete closure of the gate in these cases, thus resulting in free pressure oscillations after the complete blockage of the flow consisting mostly of noise, as opposed to the CFD and laboratory cases which exhibit significant oscillations; see Fig. 5(a–b). Since the free pressure oscillations in the full-scale case consist mostly of noise, it will not make an appreciable difference to have ξ directly proportional to dp /dx (as it approximately is for the UP) or to a weighted sum of past changes of the differential pressure gradient (as it is for the NUP).

Table 1 Summary of conditions presented in Fig. 4. LS and FS signify that the measurements were performed in laboratory scale and in full scale, respectively. Case

CFD

LS1

LS2

LS3

LS4

FS1

FS2

FS3

FS4

Re0 (×10−6) ΔT (s)

0.7

1.7

1.7

0.7

0.7

3.9

7.7

11.4

14.3

4.4

4

9

4.5

8.5

10

10

10

10

measurement data of the wall shear stress. Fig. 3(a) shows the timedevelopment of the wall shear stress for t ′ > 0 s obtained from the CFD simulation and the novel friction formulation (abbreviated NUP henceforth, shorthand for ‘new unsteady pressure-time’). The NUP-estimated wall shear stress is in excellent qualitative agreement with the CFD data at all times; however, quantitative differences owing to the simplifying assumptions exist. Specifically, the NUP estimate decreases too rapidly during the initial times, thus causing the difference between the estimate and CFD result to grow initially and attain a maximum value around t ′ ≈ 0.1 s. After passing through its maximum, the difference decreases and levels off at a near-constant value for t ′ > 0.3 s. Fig. 3(b) shows how the NUP-estimated wall shear stress compares with laboratory measurements. The measured and estimated wall shear stresses are the ensemble averages obtained from 50 realizations (see Sundstrom & Cervantes [5] for details). Note that the measurements predict a positive sign of τ at all times, which disagrees with the CFD results presented in Fig. 3(a). The positive sign of the measured wall shear stress is believed to be a measurement artefact; for, the hot-film sensors which were used to measure the wall shear stress cannot determine the sign of τ. The NUP estimate, on the other hand, predicts a negative τ; however, in order to simplify comparison, the absolute value of the estimate is plotted. Similarly to the trend observed from the CFD simulation the estimated wall shear stress exhibits a more rapid initial development than the measured τ, whereas for t ′ ≳ 0.9 s, the difference between the estimate and the measurement levels off at an approximately constant value. The quantitative differences that exist between Eq. (7) and the reference data stem from the two simplifying assumptions underpinning the derivation of the new model. Means to correct for these quantitative differences should be sought; however, such analysis is beyond the scope of the present paper.

5. Conclusions A new approach to model the frictional losses arising during a pressure-time flow rate measurement has been suggested in this work. The formulation is valid during the late stages of a pressure-time measurement, and it used in conjunction with an existing model. The idea behind the new method is to use the recently introduced ‘unsteady pressure-time’ approach (see Jonsson et al. [2]), which makes use of the so called Brunone model. In the new formulation, the Brunone model is utilized until the wall shear stress changes sign, after which the new model is interchanged in favor of the Brunone model. Two simplifying assumptions underlies the derivation of the new model, namely, i) that the Reynolds shear stress can be neglected during the times over which the model is valid, and ii) that the entire velocity distribution is zero at the instant when the new model is implemented. The resulting formulation for the wall shear stress constitutes a convolution between the pressure gradient and a weighting function, which is an equivalent formulation to that of a laminar flow decelerating from rest. Compared with presently utilized models for the friction, the novel formulation is shown to decrease the pressure-time flow rate estimation error in CFD and laboratory measurements, whereas the new formulation performs equally well as the recently introduced unsteady pressure-time formulation in full-scale measurements.

4. Performance of the NUP in flow rate calculations The performance of the new frictional formulation for pressure-time flow rate calculations is now tested for i) a CFD simulation, ii) laboratory measurements and iii) full-scale hydropower plant measurements. For the nine cases listed in Table 1, the flow rate calculation error of the NUP is compared with the SP and UP approaches, and the results are summarized in Fig. 4. The error, ε, is defined as ε = (QPT − Qref )/ Qref , in which QPT and Qref denote the pressure-time

Fig. 5. Time-developments of the differential pressure and bulk velocity in a pressure-time measurement. (a) Case FS4, (b) case LS3. In (a), Δp has been re-scaled to correspond to the same length between the pressure sensors as in (b). 5

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Acknowledgement [7]

The research presented was carried out as a part of “Swedish Hydropower Centre - SVC”. SVC has been established by the Swedish Energy Agency, Elforsk and Svenska Kraftnät together with Luleå University of Technology, The Royal Institute of Technology, Chalmers University of Technology and Uppsala University (see http://www.svc. nu).

[8] [9]

[10] [11]

Appendix A. Supplementary data

[12]

Supplementary data to this article can be found online at https:// doi.org/10.1016/j.flowmeasinst.2019.101604.

[13] [14]

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